Rho-invariant

Latest revision as of 10:51, 13 June 2013

[edit] 1 Introduction

The ${{Stub}} == Introduction == <wikitex>; The $\rho$-invariant is an invariant of odd-dimensional closed manifolds closely related to the equivariant signature. Its definition is motivated by the equivariant signature defect of even-dimensional manifolds with boundary. Namely, for manifolds with boundary the classical index formula for equivariant signature must be corrected by an additional term. It is this term which gives rise to the $\rho$-invariant. There is also another definition using bordism theory. Finally there is also an analytic defintion as a relative $\eta$-invariant. </wikitex> == Background == === G-index theorem === <wikitex>; Let $G$ be a compact Lie group acting smoothly on a smooth manifold $Y^{2d}$. The middle intersection form becomes a non-degenerate $(-1)^d$-symmetric bilinear form on which $G$ acts. The positive and negative definite subspaces are $G$-invariant and hence such a form yields an element in the representation ring $R(G)$ denoted by $\mathrm{sign_G} (Y)$. In fact $\mathrm{sign_G} (Y) \in R^{(-1)^d} (G)$ which in terms of characters means that we obtain a real (case $d$ even) / purely imaginary (case $d$ odd) character, which will be denoted as $$ \mathrm{sign_G} (-,Y) \colon g \in G \mapsto \mathrm{sign_G} (g,Y) \in \Cc. $$ The (cohomological version of the) Atiyah-Singer $G$-index theorem {{cite|Atiyah&Singer1968b|Theorem (6.12)}} tells us that if $Y$ is closed then for all $g \in G$ $$ \mathrm{sign_G} (g,Y) = L(g,Y) \in \Cc, $$ where $L(g,Y)$ is an expression obtained by evaluating certain cohomological classes on the fundamental classes of the $g$-fixed point submanifolds $Y^g$ of $Y$. In particular if the action is free then $\mathrm{sign_G} (g,Y) = 0$ if $g \neq 1$. This means that $\mathrm{sign_G} (Y)$ is a multiple of the regular representation. This theorem was generalized by Wall to topological semifree actions on topological manifolds, which is the case we will need here {{cite|Wall1999|chapter 14B}}. The assumption that $Y$ is closed is essential, and motivates the definition of the $\rho$-invariant. </wikitex> === Cobordism theory === <wikitex>; Also the result from cobordism theory is needed which says that for an odd-dimensional manifold $X$ with a finite fundamental group $G$ there always exists a $k \in \Nn$ and a manifold with boundary $(Y,\partial Y)$ such that $\pi_1 (Y) \cong \pi_1 (X) \cong G$ and $\partial Y = k \cdot X$. In other words $$ \Omega^{\textup{STOP}}_{2d-1} (BG) \otimes \Qq = 0 $$ This is due to {{cite|Conner&Floyd1964}}, {{cite|Williamson1966}} and {{cite|Madsen&Milgram1979}}. </wikitex> == Definition == === G finite === <wikitex>; {{beginthm|Definition}} \label{defn-rho-1} Let $X^{2d-1}$ be a closed manifold with $\pi_1 (X) \cong G$ a finite group. Define $$ \rho (X) = \frac{1}{k} \cdot \mathrm{sign_G} (\widetilde Y) \in \Qq R^{(-1)^d} (G)/ \langle \mathrm{reg} \rangle $$ for some $k \in \Nn$ and $(Y,\partial Y)$ such that $\pi_1 (Y) \cong \pi_1 (X)$ and $\partial Y = k \cdot X$. The symbol $\langle \textup{reg} \rangle$ denotes the ideal generated by the regular representation. {{endthm}} See {{cite|Atiyah&Singer1968b|Remark after Corollary 7.5}} for more details. Note that the manifold $Y$ in the definition always exists by the above mentioned result in cobodism theory. Furthermore the invariant is well-defined thanks to the cohomological version of the $G$-index theorem. More generally, the invariant can be defined for all closed manifolds $X^{2d-1}$ together with a homomorphism $f:\pi_1 (X)\to G$ to the finite group $G$. One uses $Y$ such that $\partial Y = k \cdot X$ and $f$ factors through $\pi_1 (Y)$, and one considers the representation on the middle homology of the induced $G$-cover of $Y$. </wikitex> === G compact Lie group === <wikitex>; {{beginthm|Definition|(Atiyah-Singer)}} \label{defn-rho-2} Let $G$ be a compact Lie group acting freely on a manifold $\widetilde{X}^{2d-1}$. Suppose in addition that there is a manifold with boundary $(Y,\partial Y)$ on which $G$ acts (not necessarily freely) and such that $\partial Y = \widetilde X$. Define $$ \rho_G (\widetilde X) \co G \ni g \mapsto \mathrm{sign_G} (g,Y) - L(g,Y) \in \Cc. $$ {{endthm}} See {{cite|Atiyah&Singer1968b|Theorem 7.4}} for more details. In this definition we think about the $\rho$-invariant as about a function $G \smallsetminus \{1\} \rightarrow \Cc$. When both definitions apply (that means when $G$ is a finite group), then they coincide, that means $\rho (X) = \rho_G (\widetilde X)$. Note that the invariant is natural under restriction from a group $G$ to a subgroup: if $H \le G$, and $G$ acts freely on $\widetilde X$, then $\rho_G (\widetilde X)|_{H\smallsetminus\{1\}}=\rho_H (\widetilde X)$. </wikitex> == References == {{#RefList:}} [[Category:Theory]] [[Category:Definitions]]\rho$-invariant is an invariant of odd-dimensional closed manifolds closely related to the equivariant signature. Its definition is motivated by the equivariant signature defect of even-dimensional manifolds with boundary. Namely, for manifolds with boundary the classical index formula for equivariant signature must be corrected by an additional term. It is this term which gives rise to the $\rho$-invariant. There is also another definition using bordism theory. Finally there is also an analytic defintion as a relative $\eta$-invariant.

[edit] 2 Background

[edit] 2.1 G-index theorem

Let $G$ be a compact Lie group acting smoothly on a smooth manifold $Y^{2d}$. The middle intersection form becomes a non-degenerate $(-1)^d$-symmetric bilinear form on which $G$ acts. The positive and negative definite subspaces are $G$-invariant and hence such a form yields an element in the representation ring $R(G)$ denoted by $\mathrm{sign_G} (Y)$. In fact $\mathrm{sign_G} (Y) \in R^{(-1)^d} (G)$ which in terms of characters means that we obtain a real (case $d$ even) / purely imaginary (case $d$ odd) character, which will be denoted as

$$\displaystyle \mathrm{sign_G} (-,Y) \colon g \in G \mapsto \mathrm{sign_G} (g,Y) \in \Cc.$$

The (cohomological version of the) Atiyah-Singer $G$-index theorem [Atiyah&Singer1968b, Theorem (6.12)] tells us that if $Y$ is closed then for all $g \in G$

$$\displaystyle \mathrm{sign_G} (g,Y) = L(g,Y) \in \Cc,$$

where $L(g,Y)$ is an expression obtained by evaluating certain cohomological classes on the fundamental classes of the $g$-fixed point submanifolds $Y^g$ of $Y$. In particular if the action is free then $\mathrm{sign_G} (g,Y) = 0$ if $g \neq 1$. This means that $\mathrm{sign_G} (Y)$ is a multiple of the regular representation. This theorem was generalized by Wall to topological semifree actions on topological manifolds, which is the case we will need here [Wall1999, chapter 14B].

The assumption that $Y$ is closed is essential, and motivates the definition of the $\rho$-invariant.

[edit] 2.2 Cobordism theory

Also the result from cobordism theory is needed which says that for an odd-dimensional manifold $X$ with a finite fundamental group $G$ there always exists a $k \in \Nn$ and a manifold with boundary $(Y,\partial Y)$ such that $\pi_1 (Y) \cong \pi_1 (X) \cong G$ and $\partial Y = k \cdot X$. In other words

$$\displaystyle \Omega^{\textup{STOP}}_{2d-1} (BG) \otimes \Qq = 0$$

This is due to [Conner&Floyd1964], [Williamson1966] and [Madsen&Milgram1979].

[edit] 3 Definition

[edit] 3.1 G finite

Definition 3.1.

Let $X^{2d-1}$ be a closed manifold with $\pi_1 (X) \cong G$ a finite group. Define

$$\displaystyle \rho (X) = \frac{1}{k} \cdot \mathrm{sign_G} (\widetilde Y) \in \Qq R^{(-1)^d} (G)/ \langle \mathrm{reg} \rangle$$

for some $k \in \Nn$ and $(Y,\partial Y)$ such that $\pi_1 (Y) \cong \pi_1 (X)$ and $\partial Y = k \cdot X$. The symbol $\langle \textup{reg} \rangle$ denotes the ideal generated by the regular representation.

See [Atiyah&Singer1968b, Remark after Corollary 7.5] for more details. Note that the manifold $Y$ in the definition always exists by the above mentioned result in cobodism theory. Furthermore the invariant is well-defined thanks to the cohomological version of the $G$-index theorem.

More generally, the invariant can be defined for all closed manifolds $X^{2d-1}$ together with a homomorphism $f:\pi_1 (X)\to G$ to the finite group $G$. One uses $Y$ such that $\partial Y = k \cdot X$ and $f$ factors through $\pi_1 (Y)$, and one considers the representation on the middle homology of the induced $G$-cover of $Y$.

[edit] 3.2 G compact Lie group

Definition 3.2 (Atiyah-Singer). Let $G$ be a compact Lie group acting freely on a manifold $\widetilde{X}^{2d-1}$. Suppose in addition that there is a manifold with boundary $(Y,\partial Y)$ on which $G$ acts (not necessarily freely) and such that $\partial Y = \widetilde X$. Define

$$\displaystyle \rho_G (\widetilde X) \co G \ni g \mapsto \mathrm{sign_G} (g,Y) - L(g,Y) \in \Cc.$$

See [Atiyah&Singer1968b, Theorem 7.4] for more details.

In this definition we think about the $\rho$-invariant as about a function $G \smallsetminus \{1\} \rightarrow \Cc$.

When both definitions apply (that means when $G$ is a finite group), then they coincide, that means $\rho (X) = \rho_G (\widetilde X)$.

Note that the invariant is natural under restriction from a group $G$ to a subgroup: if $H \le G$, and $G$ acts freely on $\widetilde X$, then $\rho_G (\widetilde X)|_{H\smallsetminus\{1\}}=\rho_H (\widetilde X)$.

4 References

• [Atiyah&Singer1968b] M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR0236952 (38 #5245) Zbl 0164.24301
• [Conner&Floyd1964] P. E. Conner and E. E. Floyd, Differentiable periodic maps, Academic Press Inc., Publishers, New York, 1964. MR0176478 (31 #750) Zbl 0417.57019
• [Madsen&Milgram1979] I. Madsen and R. J. Milgram, The classifying spaces for surgery and cobordism of manifolds, Princeton University Press, Princeton, N.J., 1979. MR548575 (81b:57014) Zbl 0446.57002
• [Wall1999] C. T. C. Wall, Surgery on compact manifolds, American Mathematical Society, Providence, RI, 1999. MR1687388 (2000a:57089) Zbl 0935.57003
• [Williamson1966] R. E. Williamson, Cobordism of combinatorial manifolds, Ann. of Math. (2) 83 (1966), 1–33. MR0184242 (32 #1715) Zbl 0137.42901